Computational Fluid Dynamics: Principles and Applications

By Jiri Blazek

Computational Fluid Dynamics: Principles and Applications, Third Edition presents students, engineers, and scientists with all they need to gain a solid understanding of the numerical methods and principles underlying modern computation techniques in fluid dynamics. By providing complete coverage of the essential knowledge required in order to write codes or understand commercial codes, the book gives the reader an overview of fundamentals and solution strategies in the early chapters before moving on to cover the details of different solution techniques.

This updated edition includes new worked programming examples, expanded coverage and recent literature regarding incompressible flows, the Discontinuous Galerkin Method, the Lattice Boltzmann Method, higher-order spatial schemes, implicit Runge-Kutta methods and parallelization.

An accompanying companion website contains the sources of 1-D and 2-D Euler and Navier-Stokes flow solvers (structured and unstructured) and grid generators, along with tools for Von Neumann stability analysis of 1-D model equations and examples of various parallelization techniques.

• Will provide you with the knowledge required to develop and understand modern flow simulation codes
• Features new worked programming examples and expanded coverage of incompressible flows, implicit Runge-Kutta methods and code parallelization, among other topics
• Includes accompanying companion website that contains the sources of 1-D and 2-D flow solvers as well as grid generators and examples of parallelization techniques

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 23, 2015
• ISBN: 9780128011720

Jiri Blazek

Jiri Blazek received his MSc in Aerospace Engineering from the Institute of Technology in Aachen, Germany in 1989. He continued his research at the German Aerospace Center, DLR, and in 1995 obtained his PhD in Aerospace Engineering, focusing on CFD methods for high-speed flows, from the University of Braunschweig, Germany. Following this, Dr. Blazek worked as a research scientist at ABB Turbosystems in Baden, Switzerland, moving to ABB Corporate Research Ltd. (now ALSTOM Power Ltd.) as researcher and project leader for CFD code development in the fields of gas and steam turbines. He was appointed as senior research scientist at the Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign, USA and in 2005 founded his own consultancy and software development firm, CFD Consulting and Analysis, in Sankt Augustin, Germany. Dr. Blazek’s main research interests include: CFD code development - especially in the area of unstructured grids, aircraft and turbomachinery aerodynamics; shape optimization; and data visualization.

Book preview

Computational Fluid Dynamics

Principles and Applications

Third Edition

Jiri Blazek, PhD

CFD Consulting & Analysis, Sankt Augustin, Germany

Table of Contents

Cover image

Title page

Copyright

Acknowledgments

List of Symbols

Subscripts

Superscripts

Abbreviations

Chapter 1: Introduction

Abstract

Chapter 2: Governing Equations

Abstract

2.1 The Flow and Its Mathematical Description

2.2 Conservation Laws

2.3 Viscous Stresses

2.4 Complete System of the Navier-Stokes Equations

Chapter 3: Principles of Solution of the Governing Equations

Abstract

3.1 Spatial Discretization

3.2 Temporal Discretization

3.3 Turbulence Modeling

3.4 Initial and Boundary Conditions

Chapter 4: Structured Finite-Volume Schemes

Abstract

4.1 Geometrical Quantities of a Control Volume

4.2 General Discretization Methodologies

4.3 Discretization of the Convective Fluxes

Chapter 5: Unstructured Finite-Volume Schemes

Abstract

5.1 Geometrical Quantities of a Control Volume

5.2 General Discretization Methodologies

5.3 Discretization of the Convective Fluxes

5.4 Discretization of the Viscous Fluxes

Chapter 6: Temporal Discretization

Abstract

6.1 Explicit Time-Stepping Schemes

6.2 Implicit Time-Stepping Schemes

6.3 Methodologies for Unsteady Flows

Chapter 7: Turbulence Modeling

Abstract

7.1 Basic Equations of Turbulence

7.2 First-Order Closures

7.3 Large-Eddy Simulation

Chapter 8: Boundary Conditions

Abstract

8.1 Concept of Dummy Cells

8.2 Solid Wall

8.3 Far-Field

8.4 Inlet/Outlet Boundary

8.5 Injection Boundary

8.6 Symmetry Plane

8.7 Coordinate Cut

8.8 Periodic Boundaries

8.9 Interface Between Grid Blocks

8.10 Flow Gradients at Boundaries of Unstructured Grids

Chapter 9: Acceleration Techniques

Abstract

9.1 Local Time-Stepping

9.2 Enthalpy Damping

9.3 Residual Smoothing

9.4 Multigrid

9.5 Preconditioning for Low Mach Numbers

9.6 Parallelization

Chapter 10: Consistency, Accuracy, and Stability

Abstract

10.1 Consistency Requirements

10.2 Accuracy of Discretization Scheme

10.3 Von Neumann Stability Analysis

Chapter 11: Principles of Grid Generation

Abstract

11.1 Structured Grids

11.2 Unstructured Grids

Chapter 12: Software Applications

Abstract

12.1 Programs for Stability Analysis

12.2 Structured 1-D Grid Generator

12.3 Structured 2-D Grid Generators

12.4 Structured to Unstructured Grid Converter

12.5 Quasi 1-D Euler Solver

12.6 Structured 2-D Euler/Navier-Stokes Solver

12.7 Unstructured 2-D Euler/Navier-Stokes Solver

12.8 Parallelization

Appendix

Abstract

Keywords

Contents

A.1 Governing Equations in Differential Form

A.2 Quasilinear Form of the Euler Equations

A.3 Mathematical Character of the Governing Equations

A.4 Navier-Stokes Equations in Rotating Frame of Reference

A.5 Navier-Stokes Equations Formulated for Moving Grids

A.6 Thin Shear Layer Approximation

A.7 PNS Equations

A.8 Axisymmetric Form of the Navier-Stokes Equations

A.9 Convective Flux Jacobian

A.10 Viscous Flux Jacobian

A.11 Transformation from Conservative to Characteristic Variables

A.12 GMRES Algorithm

A.13 Tensor Notation

Index

Copyright

Butterworth Heinemann is an imprint of Elsevier

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225 Wyman Street, Waltham, MA 02451, USA

Copyright © 2015 Elsevier Ltd. All rights reserved.

First Edition: 2005

Reprinted on: 2007

Copyright © 2005, J. Blazek. Published by Elsevier Ltd. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

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Acknowledgments

My first thanks are to our Creator, without whom nothing would be possible. Furthermore, I wish to thank my father for the initial motivation to start this project, as well as for his continuous help with the text and in particular with the drawings. I also gratefully acknowledge the support of the staff at Elsevier Ltd., foremost of C. Owen and H. Gray, during the preparation of this edition.

List of Symbols

  Jacobian of convective fluxes

  Jacobian of viscous fluxes

b  

constant depth of control volume in two dimensions

c  

speed of sound

cp   specific heat coefficient at constant pressure

cv   specific heat coefficient at constant volume

  vector of characteristic variables

Cm   molar concentration of species m (= ρYm/Wm)

CS   Smagorinsky constant

d  

distance

D  

diagonal part of implicit operator

  artificial dissipation

Dm   effective binary diffusivity of species m

e  

internal energy per unit mass

E  

total energy per unit mass

f  

Fourier symbol of the time-stepping operator

  vector of external volume forces

  flux vector

  flux tensor

g  

amplification factor

  grid velocity

h  

enthalpy

Δh   local grid (cell) size

H  

total (stagnation) enthalpy

  Hessian matrix (matrix of second derivatives)

I  

)

  identity matrix

  unit tensor

  interpolation operator

  restriction operator

  prolongation operator

  system matrix (implicit operator)

J−1   inverse of determinant of coordinate transformation Jacobian

k  

thermal conductivity coefficient

K  

turbulent kinetic energy

Kf , Kb   forward and backward reaction rate constants

lT   turbulent length scale

L  

strictly lower part of implicit operator

Lij   components of Leonard stress tensor

M  

Mach number

  mass matrix

  unit normal vector (outward pointing) of control volume face

nx , ny , nz   components of the unit normal vector in x-, y-, z-direction

N  

number of grid points, cells, or control volumes

NA   number of adjacent control volumes

NF   number of control volume faces

p  

static pressure

P  

production term of kinetic turbulent energy

  transformation matrix from primitive to conservative variables

  left and right preconditioning matrix (Krylov-subspace methods)

Pr  

Prandtl number

  heat flux due to radiation, chemical reactions, etc.

Q  

source term

  position vector (Cartesian coordinates); residual (GMRES)

  vector from point i to point j

R  

specific gas constant

Ru   universal gas constant (= 8314.34 J/kg mol K)

  residual, right-hand side

  smoothed residual

  rotation matrix

Re  

Reynolds number

  rate of change of species m due to chemical reactions

)

Sij   components of strain-rate tensor

Sx , Sy , Sz   Cartesian components of the face vector

dS   surface element

ΔS   length/area of a face of a control volume

t  

time

tT   turbulent time scale

Δt   time step

T  

static temperature

  matrix of right eigenvectors

  matrix of left eigenvectors

u, v, w   Cartesian velocity components

uτ  

U  

general (scalar) flow variable

U  

strictly upper part of implicit operator

  vector of general flow variables

  velocity vector with the components u,v, and w

V  

contravariant velocity

Vr   contravariant velocity relative to grid motion

Vt   contravariant velocity of a face of the control volume

Wm   molecular weight of species m

  vector of conservative variables (= [ρ, ρu, ρv, ρw, ρE]T)

  vector of primitive variables (= [p, u, v, w, T]T)

x, y, z   Cartesian coordinate system

Δx   cell size in x-direction

y+   nondimensional wall coordinate (= ρ yuτ/μw)

Ym   mass fraction of species m

z  

Fourier symbol of the spatial operator

α  

angle of attack, inlet angle

αm   coefficient of the Runge-Kutta scheme (in stage m)

β  

parameter to control time accuracy of an implicit scheme

βm   blending coefficient (in stage m of the Runge-Kutta scheme)

γ  

ratio of specific heat coefficients at constant pressure and volume

Γ  

circulation

  preconditioning matrix (low Mach-number flow)

δij   Kronecker symbol

ε  

rate of turbulent energy dissipation

  smoothing coefficient (implicit residual smoothing); parameter

κ  

thermal diffusivity coefficient

λ  

second viscosity coefficient

Λc   eigenvalue of convective flux Jacobian

  diagonal matrix of eigenvalues of convective flux Jacobian

  spectral radius of convective flux Jacobian

  spectral radius of viscous flux Jacobian

μ  

dynamic viscosity coefficient

ν  

kinematic viscosity coefficient (= μ/ρ)

ξ, η, ζ   curvilinear coordinate system

ρ  

density

σ  

Courant-Friedrichs-Lewy (CFL) number

σ*   CFL number due to residual smoothing

τ  

viscous stress

τw   wall shear stress

  viscous stress tensor (normal and shear stresses)

τij   components of viscous stress tensor

  components of Favre-averaged Reynolds stress tensor

  components of Reynolds stress tensor

  components of subgrid-scale stress tensor

  components of Favre-filtered subgrid-scale stress tensor

  components of subgrid-scale Reynolds stress tensor

ω  

rate of dissipation per unit turbulent kinetic energy (=ε/K)

Υ  

pressure sensor

Ω  

control volume

Ωij   components of rotation-rate tensor

∂Ω   boundary of a control volume

Ψ  

limiter function

  gradient of scalar U

∇² U   Laplace of scalar U

Subscripts

C   convective part

c   related to convection

D   diffusive part

i, j, k   nodal point index

I, J, K   index of a control volume

L   laminar; left

m   index of control volume face; species

R   right

T   turbulent

v   viscous part

V   related to volume

w   wall

x, y, z   components in the x-, y-, z-direction

  at infinity (far-field)

Superscripts

I, J, K   direction in computational space

n  

previous time level

n + 1   new time level

T  

transpose

  Favre averaged mean value; Favre-filtered value (LES)

″  

fluctuating part of Favre decomposition; subgrid scale (LES)

  Reynolds averaged mean value; filtered value (LES)

′  

fluctuating part of Reynolds decomposition; subgrid scale (LES)

Abbreviations

AGARD  

Advisory Group for Aerospace Research and Development(NATO)

AIAA  

American Institute of Aeronautics and Astronautics

ARC  

Aeronautical Research Council, UK

ASME  

The American Society of Mechanical Engineers

CERCA  

Centre de Recherche en Calcul Applique (Centre for Research on Computation and its Applications), Montreal, Canada

CERFACS  

Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique (European Centre for Research and Advanced Training in Scientific Computation), France

DFVLR  

(now DLR) Deutsche Forschungs- und Versuchsanstalt für Luft-und Raumfahrt (German Aerospace Research Establishment)

DLR  

Deutsches Zentrum für Luft- und Raumfahrt (German Aerospace Center)

ERCOFTAC  

European Research Community on Flow, Turbulence and Combustion

ESA  

European Space Agency

FFA  

Flygtekniska Försöksanstalten (The Aeronautical Research Institute of Sweden)

GAMM  

Gesellschaft für Angewandte Mathematik und Mechanik (German Society of Applied Mathematics and Mechanics)

ICASE  

Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA, USA

INRIA  

Institut National de Recherche en Informatique et en Automatique (The French National Institute for Research in Computer Science and Control)

ISABE  

International Society for Air Breathing Engines

MAE  

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NY, USA

NACA  

(now NASA) The National Advisory Committee for Aero-nautics, USA

NASA  

National Aeronautics and Space Administration, USA

NLR  

Nationaal Lucht en Ruimtevaartlaboratorium (National Aerospace Laboratory), The Netherlands

ONERA  

Office National d’Etudes et de Recherches Aerospatiales (National Institute for Aerospace Studies and Research), France

SIAM  

Society of Industrial and Applied Mathematics, USA

VKI  

Von Karman Institute for Fluid Dynamics, Belgium

ZAMM  

Zeitschrift für angewandte Mathematik und Mechanik (Journal of Applied Mathematics and Mechanics), Germany

ZFW  

Zeitschrift für Flugwissenschaften und Weltraumforschung (Journal of Aeronautics and Space Research), Germany

1D  

one dimension

1-D  

one-dimensional

2D  

two dimensions

2-D  

two-dimensional

3D  

three dimensions

3-D  

three-dimensional

Chapter 1

Introduction

Abstract

History of the computational fluid dynamics (CFD) started in the early 1970s, triggered by the availability of increasingly powerful mainframes. CFD became an acronym for the combination of physics, numerical mathematics, and, to some extent, computer sciences all employed to simulate fluid flows. Nowadays, CFD methodologies are routinely employed in the fields of aircraft, turbomachinery, car, and ship design. Furthermore, CFD is also applied in meteorology, oceanography, astrophysics, biology, oil recovery, and in architecture.

This book’s objective is to provide the university students with a solid foundation for understanding the numerical methods employed in today’s CFD and to familiarize them with modern CFD codes by providing hands-on experience. The book is also intended for engineers and scientists starting to work in the field of CFD, or who are applying CFD codes.

The book starts with the derivation of the governing equations. Chapter 3 continues with a brief description of their solution methodologies, and provides a large number of bibliographical references. Following chapters discuss in more detail about the structured and unstructured finite-volume schemes, temporal discretizations, turbulence modeling issues, application of boundary conditions, and various acceleration techniques. Further consistency, accuracy, and stability of solution methods, as well as principles of grid generation are also treated. The book closes with descriptions of the accompanying software applications, an appendix, and index.

Keywords

History of CFD

Book’s objectives

Chapter overviews

The history of the computational fluid dynamics (CFD) started in the early 1970s. Around that time, it became an acronym for the combination of physics, numerical mathematics, and, to some extent, computer sciences—all employed to simulate fluid flows. The beginning of CFD was triggered by the availability of increasingly more powerful mainframes, and still the advances in CFD are closely linked to the evolution of the computer technology. Among the first applications of the CFD methods was the simulation of transonic flows based on the solution of the non-linear potential equation. With the beginning of the 1980s, first the solutions of two-dimensional (2-D) and later three-dimensional (3-D) Euler equations became feasible. Thanks to the rapidly increasing speed of supercomputers, and due to the development of a variety of numerical acceleration techniques like multigrid, it became possible to compute inviscid flows either past complete aircraft configurations or inside of turbomachinery. With the mid-1980s, the focus started to shift to the significantly more demanding simulations of viscous flows governed by the Navier-Stokes equations. Together with this, a variety of turbulence models evolved with different degree of numerical complexity and accuracy. The leading edge in turbulence modeling is represented by the direct numerical simulation and the large eddy simulation (LES).

With the advances of the numerical methodologies, particularly of the implicit schemes, solution of flow problems that require real gas modeling also became feasible by the end of the 1980s. Among the first large scale application, 3-D hypersonic flow past re-entry vehicles, like the European HERMES shuttle, was computed using equilibrium and later non-equilibrium chemistry models. Many research activities were and still are devoted to the numerical simulation of combustion and particularly to flame modeling. These efforts are very important for the development of low emission gas turbines and engines. Also, the modeling of steam and in particular condensation of steam became a key factor in designing efficient steam turbines.

Due to the steadily increasing demands on the complexity and the fidelity of flow simulations, grid generation methods became more and more sophisticated. The development started first with relatively simple structured meshes, constructed either by algebraic methods or by using partial differential equations. But with the increasing geometrical complexity of the configurations, the grids had to be divided into a number of topologically simpler blocks (multiblock approach). The next logical step was to allow for non-matching interfaces between the grid blocks, in order to relieve the constraints imposed on the grid generation in a single block. Finally, solution methodologies were introduced that can deal with grids overlapping each other (Chimera technique). This allowed, for example, to simulate the flow past the complete Space Shuttle vehicle with the external tank and boosters attached. However, the generation of a structured, multiblock grid for a complicated geometry may still take weeks to accomplish. Therefore, the research also focused on the development of unstructured grid generators and flow solvers, which promise significantly reduced setup times, with only a minor user intervention. Another very important feature of the unstructured methodology is the possibility of solution-based grid adaptation. The first unstructured grids consisted exclusively of isotropic tetrahedra, which was fully sufficient for inviscid flows governed by the Euler equations. However, the solution of the Navier-Stokes equations at higher Reynolds numbers requires grids, which are highly stretched in the shear layers. Although such grids can also be constructed from tetrahedral elements, it is advisable to use prisms or hexahedra in the viscous flow regions and tetrahedra outside. This improves not only the solution accuracy, but it also saves the number of elements, faces, and edges. Thus, the memory and run-time requirements of the simulation are reduced significantly.

Nowadays, CFD methodologies are routinely employed in the fields of aircraft, turbomachinery, car, and ship design. Furthermore, CFD is also applied in meteorology, oceanography, astrophysics, biology, oil recovery, and in architecture. Many numerical techniques developed for CFD are also utilized in the solution of the Maxwell equations or in aeroacoustics. Hence, CFD has become an important design tool in engineering, and also an indispensable research tool in various sciences. Due to the advances in numerical solution methods and in the computer technology, geometrically and physically complex cases can be run even on PCs or on PC clusters. Large scale simulations of viscous flows on grids consisting of dozens of millions of elements can be accomplished within only a few hours on today’s supercomputers. However, it would be completely wrong to think that CFD represents a mature technology now, like, for example, the finite-element methods in solid mechanics. No, there are still many open questions like turbulence and combustion modeling, heat transfer, efficient solution techniques for viscous flows, robust but accurate discretization methods, automated grid generators, etc. The coupling between CFD and other disciplines (like the solid mechanics) requires further research as well. Quite new opportunities also arise in the design optimization by using CFD.

The objective of this book is to provide university students with a solid foundation for understanding the numerical methods employed in today’s CFD and to familiarize them with modern CFD codes by hands-on experience. The book is also intended for engineers and scientists starting to work in the field of CFD, or who are applying CFD codes. The mathematics used is always connected to the underlying physics to facilitate the understanding of the matter. The text can serve as a reference handbook too. Each chapter contains an extensive bibliography, which may form the basis for further studies.

CFD methods are concerned with the solution of equations of fluid motion as well as with the interaction of the fluid with solid bodies. The equations governing the motion of an inviscid fluid (Euler equations) and of viscous fluid (Navier-Stokes equations) are derived in Chapter 2. Additional thermodynamic relations for a perfect gas as well as for a real gas are also discussed. Chapter 3 deals with the principles of solution of the governing equations. The most important methodologies are briefly described and the corresponding references are provided. Chapter 3 can be used together with Chapter 2 to get acquainted with the fundamental principles of CFD.

Numerous schemes were developed in the past for the spatial discretization of the Euler and the Navier-Stokes equations. A unique feature of the present book is that it deals with both the structured (Chapter 4) as well as with the unstructured finite-volume schemes (Chapter 5), because of their broad application possibilities, especially for the treatment of complex flow problems routinely encountered in an industrial environment. The attention is particularly devoted to the definition of the various types of control volumes together with spatial discretization methodologies for convective and viscous fluxes. The 3-D finite-volume formulations of the most popular central and upwind schemes are presented in detail.

The methodologies for the temporal discretization of the governing equations can be divided into two main classes. One class comprises explicit time-stepping schemes (Section 6.1), and the other one consists of implicit schemes (Section 6.2). In order to provide a more complete overview, recently developed solution methods based on the Newton-iteration as well as standard techniques like the explicit Runge-Kutta schemes are discussed.

Two qualitatively different types of viscous fluid flows are encountered in general: laminar and turbulent. The solution of the Navier-Stokes equations does not raise any fundamental difficulties in the case of laminar flows. However, the simulation of turbulent flows continues to present a significant challenge as before. A relatively simple way of modeling the turbulence is offered by the so-called Reynolds-averaged Navier-Stokes equations. On the other hand, Reynolds stress models or LES enable considerably more accurate predictions of turbulent flows. In Chapter 7, various well-proven and widely applied turbulence models of varying level of complexity are presented in detail.

In order to account for the specific features of a particular problem, and to obtain an unique solution of the governing equations, it is necessary to specify appropriate boundary conditions. Basically, there are two types of boundary conditions: physical and numerical. Chapter 8 deals with both types in different situations like solid walls, inlet, outlet, injection, and far-field. Symmetry planes, periodic and block boundaries are treated as well.

In order to reduce the computer time required to solve the governing equations for complex flow problems, it is quite essential to employ numerical acceleration techniques. Chapter 9 deals extensively, among others, with approaches like the implicit residual smoothing and multigrid. Another important methodology which is also described in Chapter 9 is preconditioning of the governing equations. It allows the application of a single numerical scheme for flows, where the Mach number varies between nearly zero and transonic or higher values. Finally, Chapter 9 contains a section on the parallelization of numerical computer codes by using different approaches.

Each discretization of the governing equations introduces a certain error—the discretization error. Several consistency requirements have to be fulfilled by the discretization scheme, in order to ensure the solution of the discretized equations closely approximates the solution of the original equations. This problem is addressed in the first two parts of Chapter 10. Before a particular numerical solution method is implemented, it is important to know, at least approximately, how the method will influence the stability and the convergence behavior of the CFD code. It was frequently confirmed that the Von Neumann stability analysis can provide a good assessment of the properties of a numerical scheme. Therefore, the third part of Chapter 10 deals with stability analysis for various model equations.

One of the challenging tasks in CFD is the generation of structured or unstructured body-fitted grids around complex geometries. The grid is used to discretize the governing equations in space. The accuracy of the flow solution is therefore closely linked to the quality of the grid. In Chapter 11, the most important methodologies for the generation of structured as well as unstructured grids are discussed in depth.

In order to demonstrate the practical aspects of different numerical solution methodologies, various source codes are available for download. Provided are the sources of quasi 1-D Euler, as well as of 2-D Euler and Navier-Stokes structured and unstructured flow solvers. Furthermore, source codes of 2-D structured algebraic and elliptic grid generators are included together with a converter from structured to unstructured grids. Furthermore, two programs are provided to conduct the linear stability analysis of explicit and implicit time-stepping schemes. The source codes are completed by a set of worked out examples including the grids, the input files and the results. The code package also contains several programs for the demonstration of parallelization techniques. Chapter 12 describes the contents of the directories, the capabilities of the particular programs, and provides examples of their usage.

The present book is finalized with an Appendix and Index. The Appendix contains the governing equations presented in a differential form as well as their characteristic properties. Formulations of the governing equations in rotating frame of reference and for moving grids are discussed along with some simplified forms. Furthermore, Jacobian and transformation matrices from conservative to characteristic variables are presented for two and three dimensions. The GMRES conjugate gradient method for the solution of linear equations systems is described next. The Appendix closes with a brief explanation of the tensor notation.

Chapter 2

Governing Equations

Abstract

Computational fluid dynamics methods are concerned with the solution of equations of fluid motion as well as with the interaction of the fluid with solid bodies. Based on the conservation of mass, momentum, and energy, we derive the equations governing the motion of an inviscid fluid (Euler equations) and of a viscous fluid (Navier-Stokes equations) in their integral form. We also discuss the formulation of the viscous stress tensor. The equations are closed by the specification of thermodynamical properties of the fluid. We present appropriate formulations for perfect (ideal) gases, as well as for real gases. We conclude with a discussion of simplified governing equations, namely with thin shear-layer approximation, parabolized Navier-Stokes equations, and with Euler equations.

Keywords

Conservation laws

Euler equations

Navier-Stokes equations

Viscous stresses

Thermodynamical properties

Perfect gas

Real gas

Simplified equations

Contents

2.1  The Flow and Its Mathematical Description   7

2.1.1  Finite control volume   8

2.2  Conservation Laws   10

2.2.1  The continuity equation   10

2.2.2  The momentum equation   10

2.2.3  The energy equation   12

2.3  Viscous Stresses   14

2.4  Complete System of the Navier-Stokes Equations   16

2.4.1  Formulation for a perfect gas   19

2.4.2  Formulation for a real gas   19

2.4.3  Simplifications to the Navier-Stokes equations   23

Thin shear layer approximation   23

Parabolized Navier-Stokes equations   24

Euler equations   25

References   26

2.1 The Flow and Its Mathematical Description

Before we begin with the derivation of the basic equations describing the behavior of the fluid, it may be convenient to clarify what the term "fluid dynamics" stands for. It is, in fact, the investigation of the interactive motion of a large number of individual particles. These are in our case molecules or atoms. That means, we assume the density of the fluid is high enough, so that it can be approximated as a continuum. It implies that even an infinitesimally small (in the sense of differential calculus) element of the fluid still contains a sufficient number of particles, for which we can specify mean velocity and mean kinetic energy. In this way, we are able to define velocity, pressure, temperature, density, and other important quantities at each point of the fluid.

The derivation of the principal equations of fluid dynamics is based on the fact that the dynamical behavior of a fluid is determined by the following conservation laws, namely:

1. the conservation of mass,

2. the conservation of momentum,

3. the conservation of energy.

The conservation of a certain flow quantity means that its total variation inside an arbitrary volume can be expressed as the net effect of the amount of the quantity being transported across the boundary, of any internal forces and sources, and of external forces acting on the volume. The amount of the quantity crossing the boundary is called flux. The flux can be in general decomposed into two different parts: one due to the convective transport and the other one due to the molecular motion present in the fluid at rest. This second contribution is of a diffusive nature—it is proportional to the gradient of the quantity considered, and hence it will vanish for a homogeneous distribution.

The discussion of the conservation laws leads us quite naturally to the idea of dividing the flow field into a number of volumes and to concentrate on the modeling of the behavior of the fluid in one such finite region. For this purpose, we define the so-called finite control volume and try to develop a mathematical description of its physical properties.

2.1.1 Finite control volume

Consider a general flow field as represented by streamlines in Fig. 2.1. An arbitrary finite region of the flow, bounded by the closed surface ∂Ω and fixed in space, defines the control volume Ω. We also introduce a surface element dS . The conservation law applied to an exemplary scalar quantity per unit volume U says that its variation in time within Ω, that is,

is equal to the sum of the contributions due to the convective flux—amount of the quantity U

further due to the diffusive flux—expressed by the generalized Fick’s gradient law

where κ is the thermal diffusivity coefficient, and finally due to the volume as well as surface sources, Q V , that is,

After summing up the above contributions, we obtain the following general form of the conservation law for the scalar quantity U

  

(2.1)

where U* denotes the quantity U per unit mass, that is, U/ρ.

Figure 2.1 Definition of a finite control volume (fixed in space).

It is important to note that if the conserved quantity would be a vector instead of a scalar, Eq. (the convective flux tensorthe diffusive flux tensoras

  

(2.2)

The integral formulationof the conservation law, as given by Eqs. (2.1) or (2.2), has two very important and desirable properties:

1. If there are no volume sources present, the variation of U depends solely on the flux across the boundary ∂Ω and not on any flux inside the control volume Ω.

2. This particular form remain valid in the presence of discontinuities in the flow field like shocks or contact discontinuities [1].

Because of its generality and its desirable properties, it is not surprising that the majority of the CFD codes today is based on the integral form of the governing equations.

In the following section, we shall utilize the above integral form in order to derive the corresponding expressions for the three conservation laws of the fluid dynamics.

2.2 Conservation Laws

2.2.1 The continuity equation

If we restrict our attention to single-phase fluids, the law of mass conservation expresses the fact that mass cannot be created in such a fluid system, nor can it disappear. There is also no diffusive flux contribution to the continuity equation, since for a fluid at rest, any variation of mass would imply a displacement of the fluid particles.

In order to derive the continuity equation, consider the model of a finite control volume fixed in space, as sketched in , and dS denotes an elemental surface area. The conserved quantity in this case is the density ρ. For the time rate of change of the total mass inside the finite volume Ω we have

The mass flow of a fluid through some surface fixed in space equals to the product of (density) × (surface area) × (velocity component perpendicular to the surface). Therefore, the contribution from the convective flux across each surface element dS becomes

always points out of the control volume, we speak of inflow is negative, and of outflow if it is positive and hence the mass leaves the control volume.

As stated above, there are no volume or surface sources present. Thus, by taking into account the general formulation of Eq. (2.1), we can write

   (2.3)

This represents the integral form of the continuity equation—the mass conservation law.

2.2.2 The momentum equation

We may start the derivation of the momentum equation by recalling the particular form of Newton’s second law which states that the variation of momentum is caused by the net force acting on an mass element. For the momentum of an infinitesimally small portion of the control volume Ω (see Fig. 2.1) we have

The variation in time of momentum within the control volume equals

Hence, the conserved quantity is here the product of the density and the velocity, that is,

The convective flux tensor, which describes the transfer of momentum across the boundary of the control volume, consists in the Cartesian coordinate system of the following three components

The contribution of the convective flux tensor to the conservation of momentum is then given by

The diffusive flux is zero since there is no diffusion of momentum possible for a fluid at rest. Thus, the remaining question is now, what are the forces the fluid element is exposed to? We can identify two kinds of forces acting on the control volume:

1.  External volume or body forces, which act directly on the mass of the volume. These are, for example, gravitational, buoyancy, Coriolis, or centrifugal forces. In certain cases, there can be electromagnetic forces present as well.

2.  Surface forces, which act directly on the surface of the control volume, result from the following two sources only:

(a) the pressure distribution imposed by the fluid surrounding the volume,

(b) the shear and normal stresses resulting from the friction between the fluid and the surface of the volume.

, corresponds to the volume sources in Eq. (2.2). Thus, the contribution of the body (external) force to the momentum conservation is

The surface sources consist then of two parts—of an isotropic pressure component and of a viscous stress , that is,

   (2.4)

being the unit tensor (for tensors, see, e.g., [2]). The effect of the surface sources on the control volume is sketched in Fig. 2.2. In Section 2.3, we shall elaborate the form of the stress tensor in more detail, and in particular show how the normal and the shear stresses are connected to the flow velocity.

Figure 2.2 Surface forces acting on a surface element of the control volume.

Hence, if we now sum up all the above contributions according to the general conservation law (Eq. (2.2)), we finally obtain the expression

  

(2.5)

for the momentum conservation inside an arbitrary control volume Ω which is fixed in space.

2.2.3 The energy equation

The underlying principle that we will apply in the derivation of the energy equation is the first law of thermodynamics. Applied to the control volume displayed in Fig. 2.1, it states that any changes in time of the total energy inside the volume are caused by the rate of work of forces acting on the volume and by the net heat flux into it. The total energy per unit mass E of a fluid is obtained by adding its internal energy per unit mass, e. Thus, we can write for the total energy

  

(2.6)

The conserved quantity is in this case the total energy per unit volume, that is, ρE. Its variation in time within the volume Ω can be expressed as

Following the discussion in course of the derivation of the general conservation law (Eq. (2.1)), we can readily specify the contribution of the convective flux as

is defined for a fluid at rest, only the internal energy becomes effective and we obtain

   (2.7)

where γ = c p /c v is the ratio of specific heat coefficients and κ denotes the thermal diffusivity coefficient. The diffusion flux represents one part of the heat flux into the control volume, namely the diffusion of heat due to molecular thermal conduction—heat transfer due to temperature gradients. Therefore, Eq. (2.7) is in general written in the form of Fourier’s law of heat conduction, that is,

   (2.8)

where k is the thermal conductivity coefficientand T is the absolute static temperature.

, which we have introduced for the momentum equation, it completes the volume sources

   (2.9)

The last contribution to the conservation of energy, which we have yet to determine, are the surface sources Q S . They correspond to the time rate of work done by the pressure as well as the shear and normal stresses on the fluid element (see Fig. 2.2), that is,

   (2.10)

Sorting now all the above contributions and terms, we obtain the following expression for the energy conservation equation

  

(2.11)

The energy equation (2.11) is usually written in a slightly different form. For that purpose, we shall utilize the following general relation between the total enthalpy, the total energy and the pressure

   (2.12)

) in the energy conservation law (2.11), and apply the formula (2.12), we can write the energy equation in this final form

  

(2.13)

Herewith, we have derived integral formulations of the three conservation laws: the conservation of mass (2.3), of momentum (2.5), and of energy (2.13). In the next section, we shall work out the formulation of the normal and of the shear stresses in more detail.

2.3 Viscous Stresses

. In Cartesian coordinates its general form is given by

   (2.14)

The notation τ ij means by convention that the particular stress component affects a plane perpendicular to the i-axis, in the direction of the j-axis. The components τ xx , τ yy , and τ zz stand for the shear stresses, respectively. Figure 2.3 shows the stresses for a quadrilateral fluid element. One can notice that the normal stresses (Fig. 2.3a) try to displace the faces of the element in three mutually perpendicular directions, whereas the shear stresses (Fig. 2.3b) try to shear the element.

Figure 2.3 Normal (a) and shear stresses (b) acting on a finite fluid element.

You may ask now, how the viscous stresses are evaluated. First of all, they depend on the dynamical properties of the medium. For fluids like air or water, Newton stated that the shear stress is proportional to the velocity gradient. Therefore, medium of such a type is designated as Newtonian fluid. On the other hand, fluids like, for example, melted plastic or blood behave in a different manner—they are non-Newtonian fluids. For the vast majority of practical problems, where the fluid can be assumed to be Newtonian, the components of the viscous stress tensor are defined by the relations [3, 4]

  

(2.15)

in which λ represents the second viscosity coefficient, and μ denotes the dynamic viscosity coefficient. For convenience, we can also define the kinematic viscosity coefficient, which is given by the formula

   (2.16)

The expressions in Eq. (2.15) were derived by the Englishman George Stokes in the middle of the 19th century. The terms μ(∂u/∂x), etc. in the normal stresses represent the rate of linear in Eq. (2.15) represents volumetric dilatation—the rate of change in volume, which is in essence a change of the density.

In order to close the expressions for the normal stresses, Stokes introduced the hypothesis [5] that

   (2.17)

The above relation (2.17) is termed as the bulk viscosity. Bulk viscosity represents the property that is responsible for energy dissipation in a fluid of uniform temperature during a change in volume at finite rate.

With the exception of extremely high temperatures or pressures, there is so far no experimental evidence that Stokes’s hypothesis in Eq. (2.17) does not hold (see discussion in Ref. [6]). It is therefore generally used to eliminate λ from Eq. (2.15). Hence, we obtain for the normal viscous stresses

   (2.18)

It should be noted that the expressions for the normal stresses in Eq. ((continuity equation).

What remains to be determined are the viscosity coefficient μ and the thermal conductivity coefficient k as functions of the state of the fluid. This can be done within the framework of continuum mechanics only on the basis of empirical assumptions. We shall return to this problem in the following section.

2.4 Complete System of the Navier-Stokes Equations

In the previous sections, we have derived separately the conservation laws of mass, momentum, and energy. Now, we can collect them into one system of equations in order to obtain a better overview of the various terms involved. For this purpose, we go back to the general conservation law for a vector quantity, which is expressed in Eq. (, is related to the convective transport of quantities in the fluid. It is usually termed vector of the convective fluxes,although for the momentum and the energy equation it